Strong maximum principle for a sublinear elliptic problem at resonance

Abstract

We examine the semilinear resonant problem − Δ u = λ 1 u + λ g ( u ) in Ω , u ≥ 0 in Ω , u | ∂ Ω = 0 , where Ω ⊂ R N is a smooth, bounded domain, λ 1 is the first eigenvalue of − Δ in Ω , λ > 0 . Inspired by a previous result in literature involving power-type nonlinearities, we consider here a generic sublinear term g and single out conditions to ensure: the existence of solutions for all λ > 0 ; the validity of the strong maximum principle for sufficiently small λ . The proof rests upon variational arguments.